This paper focuses on the study of multiplicity and localized concentration properties of positive solutions for the following singularly perturbed double phase problem with nonlocal Choquard reaction {[ -ϵ ^pΔ _p u-ϵ ^qΔ _q u +V(x)(|u|^p-2u+|u|^q-2u); =ϵ ^μ -N( 1/|x|^μ*G(u)) g(u), in ℝ^N,; u∈ W^1,p(ℝ^N)∩ W^1,q(ℝ^N),u>0, in ℝ^N,; ]. where 1< p0 sufficiently small as well as related concentration properties, in relationship with the set where the potential V attains its minimum. Moreover, we also investigate the decay property of semiclassical positive solutions. The main results included in this paper complement several recent contributions to the study of concentration phenomena.

We consider a nonlinear eigenvalue problem driven by the double phase differential operator. We prove two existence theorems, both producing a continuous spectrum and the first generates eigenfunctions which blow up in the W-0(1,theta) (Omega) boolean AND L (R) (Omega)-norm ( 1 < q < p < r < q & lowast; ) , while the second generates eigenfunctions which vanish in the W-0(1,theta)(Omega) boolean AND L-infinity (Omega) -norm as lambda -> 0(+).

We investigate the existence, asymptotic boundary behavior and uniqueness of viscosity solutions u ∈ C0(Ω) of equations M_𝐚(D^2u)=f(u)+h(x) in Ω ⊂ ℝn such that u(x) → ∞ as x → ∂Ω. Such solutions are referred to as large or boundary blow-up solutions. Here, Ω is a smooth bounded domain, M_𝐚 is a weighted partial trace operator, f is a non-decreasing function that satisfies the Keller–Osserman condition, and h is a continuous function in Ω. The main difficulty in the investigation rests on the possibility that M_𝐚 is very degenerate elliptic, and h is unbounded as well as sign-changing in Ω. To the best of our knowledge, large solutions to equations involving partial trace operators have not been investigated before.

We consider a Dirichlet problem driven by the nonautonomous, degenerate p-Laplacian, with a reaction resonant at +/-infinity and at 0. Using variational tools and critical groups (Morse theory), we show that the problem has at least two nontrivial bounded solutions. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we study multivalued nonlocal elliptic problems driven by the fractional double phase operator with variable exponents and ω -logarithmic perturbation formulated by {[ ( -Δ) ^s_ℋ u ∈ℱ(x,u) in Ω ,; u=0 on ℝ^N∖Ω . ]. We are going to establish maximum principles for the fractional perturbed double phase operator and show the boundedness of weak solutions to the above problem. Finally, under appropriate assumptions we discuss the existence of infinitely many small (non-negative) weak solutions to a single-valued nonlocal double phase problem.

This paper is concerned with the existence and symmetry breaking properties of vectorial ground states for the following fractional Hartree–Fock system with a general potential in ℝ^2 : {[ (-Δ )^su+u+l(x)ϕ ^t_l,(u,v)u=( |u|^2p+|v|^p+1|u|^p-1) u in ℝ^2,; (-Δ )^sv+v+l(x)ϕ ^t_l,(u,v)v=( |v|^2p+|u|^p+1|v|^p-1) v in ℝ^2 , ]. where 1/2

This paper investigates the qualitative properties of normalized solutions to the upper critical Choquard equation with nonlocal perturbation: {[ -Δ u+λ u=(I_α *|u|^N+α/N-2)|u|^N+α/N-2-2u+μ (I_β *|u|^p)|u|^p-2u, x∈ℝ^N,; ∫ _ℝ^Nu^2dx=c, ]. where N ≥ 3 , α ,β∈ (0,N) , p ∈( N+β/N, N+β/N-2) , μ∈ℝ , c>0 , λ∈ℝ is an unknown Lagrange multiplier, and I_α ,I_β denote the Riesz potentials. For μ > 0 , we establish the existence of normalized solutions in several regimes, that is, when N+β/N< p < N+β +2/N (mass-subcritical), p = N+β +2/N (mass-critical), and N+β +2/N< p < N+β/N-2 (mass-supercritical). For μ≤ 0 , we derive a non-existence result. Particularly, to obtain sharp energy estimates crucial for restoring compactness, we classify analyses by ranges of α , β across different dimensions, developing tailored scaling techniques within each range to control energy levels below the corresponding compactness thresholds. This enables us to resolve open problems in sharp energy estimation for the mass-subcritical regime: we cover full parameter ranges for N=3,4 and extend admissible parameter ranges for N ≥ 5 , while providing a more comprehensive characterization of α , β to advance related research. Moreover, the framework applies directly to special cases including α = β and van der Waals-type potentials ( p = N+α/N-2 with α < β ), improving upon existing literature in these settings. We anticipate that the energy estimation techniques introduced in this paper will be extended to wider classes of nonlocal critical elliptic equations with mass constraint.

We study a nonlinear Dirichlet problem eigenvalue driven by a differential operator with unbalanced growth (double phase problem) and a reaction that has the competing effects of a singular term and of a superlinear perturbation. We prove an existence and multiplicity theorem which is global in the parameter lambda > 0.

This paper is concerned with the existence and multiplicity of normalized solutions to the following Schrödinger–Poisson system: {[ -ε ^2Δ u + V(x) u - ϕ |u|^3 u = λ u + μ |u|^q-2 u + |u|^4 u, in ℝ^3,; -ε ^2Δϕ = |u|^5, in ℝ^3, ]. with prescribed mass ∫ _ℝ^3 |u|^2 dx = a^2ε ^3, where a > 0 , μ > 0 , q ∈( 2, 10/3) , and ε > 0 is a small parameter. Here, λ∈ℝ arises as a Lagrange multiplier, and the potential V: ℝ^3→ [0, +∞ ) is a continuous function satisfying suitable conditions. By combining truncation techniques with some adequate estimates, we establish that, for sufficiently small ε > 0 , normalized solutions do exist. Moreover, by employing Ljusternik-Schnirelmann theory, we find a relationship between the number of positive solutions and the topology of the set where the potential V attains its minimum. Our work extends and complements recent contributions of X. Feng [17, 18] (Z. Angew. Math. Phys. 2020), to the abstract setting of multiple normalized concentrating solutions. This study seems to be the first work dealing with the existence of multiple normalized semiclassical states for the Sobolev critical Schrödinger–Poisson system coupled with a nonlocal critical term in the whole space ℝ^3 .

Abstract In this paper, we investigate the existence of normalized ground state solutions for the following mixed local and nonlocal Laplacian with Trudinger–Moser nonlinearity and singular weight { ( − Δ ) n / α α v − μ Δ p v = λ | v | n α − 2 v + f ( v ) | x | β , x ∈ R n , ∫ R n | v | n α d x = m , where n ⩾ 2 , Δ p denotes the p -Laplacian with p ⩾ n , ( − Δ ) n / α α represents the fractional n / α − Laplacian with α ∈ ( 0 , 1 ) , μ ⩾ 0 , 0 < β < n , m > 0 , λ ∈ R acts as a Lagrange multiplier, f ∈ C ( R , R ) satisfies the exponential critical growth. We first establish a Trudinger–Moser inequality with weights in fractional Sobolev spaces. Then under some general assumptions and without symmetry constraints, the existence of mountain pass type solutions is obtained. Finally, we show that the mountain pass type solution is a ground state. Key innovations include: 1. Our approach completely avoids radial symmetry assumptions. 2. We address challenges from both the weighted singularity and critical Trudinger–Moser growth. 3. New techniques are developed to circumvent the absence of Pohozaev-type identities. These results are novel even for the classical case n = 2 , significantly extending existing studies on normalized solutions for mixed local and nonlocal operators.

This paper is concerned with the study of elliptic differential problems involving fractional variable exponent double phase operators with logarithmic perturbation (-\Delta)s \scrH generated by \scrH(x, y, t) = [tp(x,y) p(x,y) +\mu(x, y) tq(x,y) q(x,y) ] log(e+\alphat). In the first part, we study fractional double phase elliptic inclusions with a generalized multivalued mapping and a maximal monotone operator which is formulated by the convex subdifferential of the indicator function to a convex set. Based on the subsupersolution method along with truncation techniques and nonsmooth analysis we show an existence result and give an application construction such a pair of sub-supersolution. Additionally, under lattice conditions, we establish the compactness and the directedness of the solution set within a pair of suband supersolutions. In the second part, we consider a type of fractional Kirchhoff double phase problems governed by the operator (-\Delta)s\scrH. Applying variational methods, the Poincare'\--Miranda existence theorem together with the quantitative deformation lemma, we prove a multiplicity result which says that the problem has at least a positive solution, a negative solution, and a sign-changing solution.

In this paper, we consider the existence of solutions for Choquard equation of the form -Δ u+V(|x|) u =[I_α *(Q(|x|)F(u))]Q(|x|)f(u), x∈ℝ^2, where the nonlinear term f has exponential growth, the radial potentials V, Q: ℝ^+→ℝ are unbounded, singular at the origin or decaying to zero. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weaker assumptions. Our study extends and improves the results of [Albuquerque-Ferreira-Severo, Milan J. Math. 89 (2021)] and [Alves-Shen, J. Differential Equations, 344 (2023)].

We consider a nonlinear Dirichlet problem with gradient dependence. The features of this paper are twofold: (i) the problem is driven by a general nonlinear nonhomogeneous differential operator with Uhlenbeck-Lieberman structure; (ii) the reaction blows-up at the origin and it is gradient dependent. Using a topological approach based on fixed point theory, we show that for all small values of lambda > 0 there are "eigenvalues" of the problem with smooth corresponding eigenfunctions.

We examine three singular Dirichlet problems driven by the double phase operator. One of the problems is nonparametric and the other two are parametric. In all problems, the perturbation is “superlinear”, but does not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity results for the problems. For the parametric problems, the results are global in the parameter λ >0 . Our approach uses variational tools from the critical point theory, truncations and comparisons and critical groups.

In this paper, we investigate the normalized solutions of a class of quasilinear elliptic equations characterized by convolution nonlinearity. We establish the existence and nonexistence of global and local minimizers across various ranges of the exponent, thereby extending many existing results in the literature.

We consider a nonlinear Robin problem driven by a differential operator with unbalanced growth and a reaction which exhibits the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) term. Using the Nehari method, we show that for all small values of the parameter, the problem has two bounded, ground state (least energy) solutions. In the process of the proof, we establish some auxiliary results which are of independent interest.

We are interested in the existence of positive bound solutions for the following fractional Choquard equation: { (-Delta)(s)u+v(x)u=(integral(|u(y)2*mu,s/)(Omega)(|x-y|)mu dy)(|u|2* mu,s-2 u, x is an element of Omega,) where Omega subset of N-& Ropf; is an unbounded exterior domain, partial derivative Omega not equal & empty;, & Ropf;(N)\Omega is bounded, s is an element of(0,1), N >2s, 0

In this paper, we establish concentration and multiplicity properties of positive ground state solutions to the following perturbed pseudo-relativistic Schrodinger equation with competing potentials {(-epsilon(2)Delta + m(2))(s)u+ V (x)u = K(x)f(u) in R-N, u is an element of H-s (R-N), u> 0 in R-N, where N > 2s, epsilon is a small positive parameter, and (-Delta + m(2))(s) is the pseudo-relativistic Schrodinger operator with s 2 (0,1) and mass m> 0. We assume that the potentials V, K and the nonlinearity f are continuous but are not necessarily of class C 1. Under natural hypotheses, combining the extension method, Nehari analysis and the Ljusternik-Schnirelmann category theory, we first study the existence and concentration phenomena of positive solutions for epsilon > 0 sufficiently small, as well as multiplicity properties depending on the topology of the set where V attains its global minimum and K attains its global maximum. Moreover, we establish the asymptotic convergence and the exponential decay of positive solutions. In the final part of this paper, we provide a sufficient condition for the non-existence of ground state solutions.